Unified Framework of Distributional Regret in Multi-Armed Bandits and Reinforcement Learning

TL;DR

This paper introduces a unified probabilistic framework for distributional regret in multi-armed bandits and reinforcement learning, providing bounds that balance expected performance and tail risk.

Contribution

It proposes a simple UCBVI-style algorithm with adjustable exploration bonuses and derives optimal distributional regret bounds in various regimes.

Findings

Achieves distributional regret bounds of order O(√(AT) log(1/δ)) for multi-armed bandits.

Provides a unified framework for gap-independent and gap-dependent bounds.

Confirms a conjecture by Lattimore & Szepesvári (2020) regarding distributional regret.

Abstract

We study the distribution of regret in stochastic multi-armed bandits and episodic reinforcement learning through a unified framework. We formalize a distributional regret bound as a probabilistic guarantee that holds uniformly over all confidence levels $\delta \in (0,1]$, thereby characterizing the regret distribution across the full range of $\delta$. We present a simple UCBVI-style algorithm with exploration bonus $\min\{c_{1,k}/N, c_{2,k}/\sqrt{N}\}$, where $N$ denotes the visit count and $(c_{1,k},c_{2,k})$ are user-specified parameters. For arbitrary parameter sequences, we derive general gap-independent and gap-dependent distributional regret bounds, yielding a principled characterization of how the parameters control the trade-off between expected performance, tail risk, and instance-dependent behavior. In particular, our bounds achieve optimal trade-offs between expected and distributional regret in both minimax and instance-dependent regimes. As a special case, for multi-armed bandits with $A$ arms and horizon $T$, we obtain a distributional regret bound of order $\mathcal{O}(\sqrt{AT}\log(1/\delta))$, confirming the conjecture of Lattimore & Szepesv\'ari (2020, Section 17.1) for the first time.